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The Curtis-Wellington spectral sequence through cohomology

Dana Hunter

Abstract

We study stable homotopy through unstable methods applied to its representing infinite loop space, as pioneered by Curtis and Wellington. Using cohomology instead of homology, we find a width filtration whose subquotients are simple quotients of Dickson algebras. We make initial calculations and determine towers in the resulting width spectral sequence. We also make calculations related to the image of $J$ and conjecture that it is captured exactly by the lowest filtration in the width spectral sequence.

The Curtis-Wellington spectral sequence through cohomology

Abstract

We study stable homotopy through unstable methods applied to its representing infinite loop space, as pioneered by Curtis and Wellington. Using cohomology instead of homology, we find a width filtration whose subquotients are simple quotients of Dickson algebras. We make initial calculations and determine towers in the resulting width spectral sequence. We also make calculations related to the image of and conjecture that it is captured exactly by the lowest filtration in the width spectral sequence.

Paper Structure

This paper contains 12 sections, 16 theorems, 60 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Background
  4. Plan of the paper
  5. Thanks
  6. Review
  7. Width spectral sequence
  8. $h_0$ towers
  9. First analysis of J map
  10. Appendix
  11. The width spectral sequence.
  12. Pairing Matrices for the homology and cohomology of $SO$

Key Result

Theorem 1.1

There is a width filtration (see Definition D:widthFiltration below) of the indecomposables of $H^*(Q_0S^0, {\mathbb F}_2)$ whose subquotients are $D_n^o$.

Figures (5)

  • Figure 1: Skyline diagram for $\gamma_1^3 \odot \gamma_2\gamma_{1[2]}^3 \odot \gamma_{2[2]}\gamma_{1[4]}^2$
  • Figure 2: Skyline diagrams for the three summands of ${Sq}^3(\gamma_{2[4]})$
  • Figure 3: The $E_1$ page of the width spectral sequence, with width filtration encoded by color: black corresponds to $D_1$, red to $D_2$, green to $D_3$, teal to $D_4$, and purple to $D_5$.
  • Figure 4: Diagram of $\mathfrak{N}$ as an unstable $\mathcal{A}$ module.
  • Figure 5: $E_1$ page of the width spectral sequence. Black corresponds to $D_1$, red to $D_2$, green to $D_3$, teal to $D_4$, purple to $D_5$, and brown to $D_6$.

Theorems & Definitions (28)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Conjecture 1.6
  • Theorem 2.1: GSS
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4: Theorem 8.3 of GSS
  • ...and 18 more